Learn Spanish (Latin America): La logia respetará el juramento mientras tengas el emblema. - The lodge will respect the oath as long as you have the emblem.

Because the action refers to an ongoing future condition. When mientras points to a future timeframe that hasn't finished yet, it requires the present subjunctive. If you used the indicative tienes, it would just mean a factual "while you (currently) have the emblem."

I usually translate mientras as "while." Why does it mean "as long as" here?

When mientras is followed by the subjunctive to talk about the future, it creates a conditional meaning. The lodge respecting the oath depends entirely on you holding the emblem. Translating it as "as long as" or "provided that" captures this requirement much better than "while."

Why is it el emblema if the word ends in "-a"?

Emblema is a masculine word despite its ending. Many Spanish nouns ending in "-ma" that originated from Greek are masculine. Other common examples you likely know include el problema, el sistema, and el clima.

The first half uses the future tense (respetará). Could I use the present tense there instead?

Yes, you could say La logia respeta..., but it slightly changes the context. Using the present tense makes it sound like a general, established rule of the organization. Using the future tense (respetará) makes it a specific promise or warning directed at the person currently holding the emblem.